4.7 Inverse Trig Functions
Inverse trig functions
• What trig functions can we evaluate without using a
calculator?
– Sin
–
𝜋
4
𝜋
Cos
3
– Tan
𝜋
6
– Sin
𝜋
2
Inverse Trig Functions
• What does an inverse function do?
– Finds the input of a function when given the output
• How can we determine if a function has an
inverse?
– Horizontal Line Test
– If any horizontal line intersects the graph of a function
in more than one point, the function does not have an
inverse
Does the Sine function have an
inverse?
1
-1
What could we restrict the domain to
so that the sine function does have an
  
inverse?
 ,
 2
1
-1
2 
-1
Inverse Sine, Sin (x) , arcsine (x)
•
•
•
•
•
Function is increasing
Takes on full range of values
Function is 1-1
Domain:  1, 1
Range:   ,  
 2
2
Evaluate: arcSin
3
2
3
• Asking the sine of what angle is
2
Find the following:
2
a) ArcSin
2
1
b) Sin ( )
2
-1
3
c) ArcSin 
2
Inverse Cosine Function
• What can we restrict the domain of the cosine
curve to so that it is 1-1? 0 ,  
1
-1
-1
Inverse Cosine, Cos (x) , arcCos (x)
•
•
•
•
•
Function is increasing
Takes on full range of values
Function is 1-1
Domain:  1 , 1
Range:   ,  
 2
2
Evaluate: ArcCos (-1)
• The Cosine of what angle is -1?
Evaluate the following:
a) Cos -1 ( 3 )
2
1
b) ArcCos (- )
2
c) Cos -1 (- 2 )
2
arcCos (0.28)
• Is the value 0.28 on either triangle or curve?
• Use your calculator:
–
-1
Cos (0.28)
Determine the missing Coordinate
Determine the missing Coordinate
Use an inverse trig function to write θ
as a function of x.
2x
θ
x+3
Find the exact value of the expression.
Sin [ ArcCos
 2
 
 3
]
4.7 Inverse Trig Functions
So far we have:
1) Restricted the domain of trig functions to
find their inverse
2) Evaluated inverse trig functions for exact
values
ArcTan (x)
• Similar to the ArcSin (x)
• Domain of Tan Function:
• Range of Tan Function:
Composition of Functions
From Algebra II:
If two functions, f(x) and f −1 (x), are inverses,
then their compositions are:
f(f −1 (x)) = x
and f −1 (f(x)) = x
Inverse Properties of Trig Functions
𝜋
2
• If -1 ≤ x ≤ 1 and - ≤ y ≤
𝜋
,
2
then
Sin (arcSin x) = x and arcSin (Sin y) = y
• If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then
Cos (arcCos x) = x and arcCos (Cos y) = y
𝜋
2
• If x is a real number and - < y <
𝜋
,
2
then
Tan (arcTan x) = x and arcTan (Tan y) = y
Inverse Trig Functions
• Use the properties to evaluate the following
expression:
Sin (ArcSin 0.3)
Inverse Trig Functions
• Use the properties to evaluate the following
expression:
ArcCos (Cos
2𝜋
)
3
Inverse Trig Functions
• Use the properties to evaluate the following
expression:
ArcSin (Sin 3π)
Inverse Trig Functions
• Use the properties to evaluate the following
expression:
a) Tan (ArcTan 25)
b) Cos (ArcCos -0.2)
c) ArcCos (Cos
7𝜋
)
2
4.7 Inverse Trig Functions
Inverse Trig Functions
• Yesterday, we only had compositions of
functions that were inverses
• When we have a composition of two functions
that are not inverses, we cannot use the
properties
• In these cases, we will draw a triangle
Inverse Trig Functions
3
4
• Sin (arcTan )
– Let u = whatever is in parentheses
• u = arcTan
3
4
→ Tan u =
3
4
Inverse Trig Functions
5
7
• Sec (arcSin )
Inverse Trig Functions
4
5
• Sec (arcSin )
5
8
• Cot (arcTan - )
• Sin (arcTan x)
Inverse Trig Functions
• In this section, we have:
– Defined our inverse trig functions for specific
domains and ranges
– Evaluated inverse trig functions
– Evaluated compositions of trig functions
• 2 Functions that are inverses
• 2 Functions that are not inverses by evaluating the
inner most function first
• 2 Functions that are not inverses by drawing a triangle
Sine Function
1
-
𝜋
2
𝜋
2
-1
Cosine Function
1
𝜋
2
-1
π
Tangent Function
-
𝜋
2
𝜋
2
Evaluating Inverse Trig Functions
a) arcTan (- 3 )
b) Cos
−1
(−
c) arcSin (-1)
3
2
)
Composition of Functions
• When the two functions are inverses:
a) Sin (arcSin -0.35)
b) arcCos (Cos
3𝜋
4
)
Composition of Functions
• When the two functions are not inverses:
−1
a) Sin
(Cos
b) arcTan (Sin
11π
)
6
4𝜋
3
)
Composition of Functions
• When the two functions are not inverses:
3
5
a) Sin (arcCos )
b) Cot
−1 12
(Sin
13
)
4.7 Inverse Trig Functions
Inverse Trig Functions
• Evaluate the following function:
f(x) = Sin (arcTan 2x)
In your graphing calculator, graph both of these
functions.
Inverse Trig Functions
• Solve the following equation for the missing
piece:
9
x
arcTan = arcSin (___)
Inverse Trig Functions
• Find the missing pieces in the following
equations:
a) arcSin
b) arcCos
c) arcCos
36−x2
6
= arcCos (___)
3
x2 −2x+10
x −2
2
= arcSin (___)
= arcTan (___)
Inverse Trig Functions
Inverse Trig Functions
Composition of Functions
1) Evaluate innermost function first
2) Substitute in that value
3) Evaluate outermost function
1
Sin (arcCos )
2
Evaluate the innermost function first:
arcCos ½ =
Substitute that value in original problem

Sin
3
7 

Cos  Sin

6 

-1

-1 5 
Tan  Cos

13 

How do we evaluate this?
Let θ equal what is in parentheses
5
  Cos
13
-1
5
 Cos 
13
5
Cos 
13
13
θ
5
12

-1 5 
Tan  Cos
  Tan 
13 

How do we evaluate this?
Let θ equal what is in parentheses
Use the triangle to answer the question
12
Tan  
5
13
θ
5
12

-1 15 
Csc Tan - 
8


-1
Sin Sin 0.2

What is different about this problem?
Is 0.2 in the domain of the arcSin?


Then Sin Sin 0.2  0.2
-1
4 

Sin  Sin

3 

-1
What is different about this problem?
4
Is
in the domain of the Sin function?
3
4
Since it is not, we must evaluate Sin
3
Graph of the ArcSin
Y



2
3
6
X = Sin Y
1
1
 3
0



6
3
2
2
2
0
1
2
3
2
1
Graph of the ArcSin
Graph of ArcCos
Y
0


6
X = Sin Y
1
3
1
2
3
2
2
0

2
3
5

6
1
2
 3
2
1
Graph of the ArcCos